Difference between revisions of "1986 AHSME Problems/Problem 11"
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In a right triangle, the length of the hypotenuse is twice the length of the median which bisects it. If the hypotenuse is <math>13</math>, the median must be <math>6.5</math>. | In a right triangle, the length of the hypotenuse is twice the length of the median which bisects it. If the hypotenuse is <math>13</math>, the median must be <math>6.5</math>. | ||
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+ | == Solution 2 (Self Torture) == | ||
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+ | Warning: this solution is very intensive in calculation. Please do NOT try this on the test! | ||
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+ | Let's start by finding <math>AH</math>. By Heron's Formula, <math>s=\frac{13+14+15}{2}=21, [ABC]=\sqrt{21*(21-13)(21-14)(21-15)}=84</math>. Using the area formula <math>A=0.5bh</math>, <math>AH=12</math>. Now using the Pythagorean Theorem, <math>BH=5, HC=9</math>. | ||
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+ | Now <math>AM=MB=6.5</math>. Using Stewart's Theorem on <math>\triangle{ABH}</math>, letting <math>HM=x</math>: | ||
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+ | <math>13*6.5*6.5+13x^2=12*6.5*12+5*6.5*5</math> (<math>man+dad=bmb+cnc</math>). | ||
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+ | Thus <math>x=6.5</math> or <math>x=-6.5</math> (reject this solution since <math>x</math> is positive). Thus <math>HM=6.5</math>. Select <math>\boxed{B}</math>. | ||
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+ | ~hastapasta | ||
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+ | P.S.: Although this is torturous, this is a good practice of Heron's formula and Stewart's theorem though. | ||
== See also == | == See also == |
Revision as of 18:58, 30 November 2022
Contents
[hide]Problem
In and . Also, is the midpoint of side and is the foot of the altitude from to . The length of is
Solution
In a right triangle, the length of the hypotenuse is twice the length of the median which bisects it. If the hypotenuse is , the median must be .
Solution 2 (Self Torture)
Warning: this solution is very intensive in calculation. Please do NOT try this on the test!
Let's start by finding . By Heron's Formula, . Using the area formula , . Now using the Pythagorean Theorem, .
Now . Using Stewart's Theorem on , letting :
().
Thus or (reject this solution since is positive). Thus . Select .
~hastapasta
P.S.: Although this is torturous, this is a good practice of Heron's formula and Stewart's theorem though.
See also
1986 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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